The use of composite materials in all types of engineering structures has led to an increased interest in the theory, analysis, design and manufacturing of structural components made of composite materials. The last few decades have seen a major effort to develop composite material systems and analyze and design structural components made from them.
The analysis of natural frequencies of composite plates/shells plays an increasingly important role in the design of structures in mechanical, civil and aerospace engineering applications. A thorough study of the dynamic behaviors of these structures is essential in assessing their full potential. Therefore, it is necessary to develop appropriate models capable of accurately predicting their dynamic characteristics.
Great progress has been made over past decades towards better understanding
of the vibration characteristics of laminated composite plates/shells (Bert,
1984; Chen et al., 1989; Cheung
and Kwok, 1975). Due to limited availability of analytic solutions for practical
applications, numerical approximate methods have become the most effective tools.
The Finite Element Method (FEM) is considered to be a very effective and versatile
approach for these problems. There is a vast amount of literature on free vibration
analysis of laminated plates/shells which is too large to list here.
Bert (1984) have conducted surveys and provided details
on the development of the FEMs for modeling and modal analysis of laminated
plates/shells. Further extensive references on shells can be found in the excellent
review of Zhang and Yang (2009).
This review contain vibration analysis of composite plates, buckling and post buckling analysis and optimization in composite plates.
In this study, we have discussed the free vibration analysis of composite laminated plates.
The first-order shear deformation theory (FSDT) has been employed widely to establish finite element models for free vibration analysis of the composite laminated plates. The effects of lamination and extension-bending coupling, shear and twist-curvature couplings on the lowest frequencies and corresponding mode shapes for free vibration of laminated anisotropic composite plates was investigated using a finite element method with quadratic interpolation functions and five engineering Degrees of Freedom (DOF). The free and forced vibration response of laminated composite folded plate structures was predicted by a nine-node Lagrangian plate-bending finite element with five engineering DOF per node that incorporated rotary inertia.
In this study, we discuss Buckling and postbuckling analysis of laminated composite
plates. The buckling of laminated composite plates is an important consideration
in the design process; however the critical value of load given by linear buckling
analysis may not accurately represent the load-carrying capability of a plate.
Although composite laminated plates generally possess less load-carrying capacity
after buckling compared to their metallic counterparts, the total load during
the postbuckling of a composite laminated plate is still several times that
of the critical buckling load. In order to get the practical limits of the load-carrying
capability of the composite laminated plates, the postbuckling behavior has
been studied to establish the sustained additional loads after buckling. Considerable
efforts have been made for the numerical analysis of the buckling and postbuckling
analysis over the years.
In this study, we discuss Failure analysis. Under normal operating conditions, local failures such as matrix cracks, fibre breakage, fibre matrix debonding and inter-layer delamination, may be developed in the laminated composite structures and the failure may cause permanent loss of integrity within the laminate and result in loss of stiffness and strength of the material. Prediction of the failure process, the initiation and growth of the damages and the maximum loads that the structures can withstand before failure occurs is essential for assessing the performance of composite laminated plates and for developing reliable and safe design. In particular, the first-ply failure analysis of laminated composite plates has been actively investigated in recent years and the mechanical behaviour and the first-ply failure load of laminated composite plates subjected to in-plane loading conditions, such as tension, compression, shear and out-of-plane loading such as transverse loads have been studied.
Free vibrational characteristics of layered circular plates are considered
by Venkatesan and Kunukkasseril (1978). The equations
incorporating shear deformation and rotatory inertia are developed for the asymmetric
motion. For axisymmetric motion, exact closed form solutions are obtained. Timothy
and Nayfeh (1996) developed the analysis and numerical calculations for
the exact free vibration characteristics of simply supported, rectangular, thick,
multilayered composite plates and assumed that each layer of the composite plate
is of arbitrary thickness, is perfectly bonded to adjacent layers, possesses
up to orthotropic material symmetry and that its material crystallographic axes
are oriented either parallel or perpendicular to the plates boundaries.
Exact formal solutions are obtained for the individual layers which are, in
turn, used to relate the field variables at the upper and lower layer surfaces.
The solution is carried through by the successive application of appropriate
interfacial continuity conditions between adjacent lamina.
The free vibration analysis of mass-loaded rectangular composite laminates
plate with mixed boundaries was performed by Chang and Wu
(1997) by using the orthogonal polynomial functions and Ritz method and
developed the subdomain method to derive the governing eigenvalue equation.
In the solution process, we used the subdomain weighted residual to satisfy
the compatibility at the interconnect boundaries for two adjacent subdomain
and carry out continuity matrices, then we adopted the Gram-Schmidt orthogonalization
process to find the orthogonal functions set which satisfy the simply subdomain
boundary condition. Finally, used the continuous matrices to develop the global
energy functional and applied the Ritz method to obtain the governing eigenvalue
equation. By solving the governing eigenvalue equation, the natural frequencies
and mode shapes of the composite laminates are obtained. Axisymmetric Vibrations
of Orthotropic Composite Circular Plates is analysed by Greenberg
and Stavsky (1978). In this a sixth order system of equations of motion
is formulated in terms of the radial and transverse displacements for axisymmetric
vibrations of circular plates laminated of polar orthotropic plies. Previous
results for heterogeneous isotropic circular plates are included as a special
case in the present theory.
A finite element model is developed by Liu et al.
(1999) for the shape control and active vibration suppression of laminated
composite plates with integrated piezoelectric sensors and actuators. The model
is based on the classical laminated plate theory and the principle of virtual
displacements. Four-node rectangular Non-conforming plate bending elements are
used to model the laminated composite plate. A computationally efficient and
highly accurate numerical method is proposed by Cheung and
Zhou (2001) to analyze the vibrations of symmetrically laminated rectangular
composite plates with intermediate line supports. A set of admissible functions
are developed from the static solutions of a beam with intermediate point supports
under a series of sinusoidal loads, in which the beam may be considered to be
a unit width strip taken from the plate in a direction parallel to the edges
of the plate. In addition to satisfying both the geometric boundary conditions
of the plate and the zero deflection conditions at the line supports, this set
of static beam functions, being different from the existing admissible functions,
can also properly describe the discontinuity of the shear forces at the line
supports, so that more accurate results can be expected for the dynamic analysis
of laminated rectangular plates with intermediate line supports. The governing
eigen frequency equation of the plates is derived by using the Rayleigh-Ritz
A meshless approach based on the reproducing kernel particle method is developed
for the flexural, free vibration and buckling analysis of laminated composite
plates. In this approach, the first-order shear deformation theory (FSDT) is
employed and the displacement shape functions are constructed using the reproducing
kernel approximation satisfying the consistency conditions. The essential boundary
conditions are enforced by a singular kernel method. Numerical examples involving
various boundary conditions are solved by Wang et al.
(2002) to demonstrate the validity of the proposed method. Comparison of
results with the exact and other known solutions in the literature suggests
that the meshless approach yields an effective solution method for laminated
The dynamic performance of a multi-layered composite plate with embedded shape
memory alloy (SMA) wires has been investigated in terms of the changes in its
relative fundamental natural frequency by Arkadiusz et
al. (2003). A sensitivity analysis has been carried out on the influence
of various geometrical parameters and material properties on the plates
dynamic performance, as well as the influence of the form of boundary condition.
The use of the Active Property Tuning (APT) method and the Active Strain Energy
Tuning (ASET) method has also been discussed within the study. The finite element
method has been used for the analysis and a new element has been exploited for
modelling multi-layered composite plates. It has been found that the dynamic
performance of the multi-layered composite plate with embedded SMA wires strongly
depends on the plate geometry and the form of boundary condition; however, the
dynamics can be successfully controlled and influenced by an optimal selection
of the geometrical parameters and material properties.
Aydogdu and Timarci (2003) has studied the vibration
analysis of cross-ply laminated square plates subjected to different sets of
boundary conditions. The analysis is based on a five-degree-of-freedom shear
deformable plate theory. The requirement of the continuity conditions among
the layers for the symmetric cross-ply laminated plates are fulfilled by the
use of the shape functions incorporated into this theory which, also, unifies
the two-dimensional shear deformable plate theories developed previously. Initially,
the governing equations obtained by use of Hamiltons principle for the
vibration of cross-ply laminated plates with simply supported boundary conditions
at all of their edges are solved by an exact analytical method.
A finite element model, based on third-order shear deformation theory, is used
in this study by Latheswary et al. (2004a) studied
the linear and the non-linear free vibration analysis of laminated composite
plates. This study has been motivated by the lack of open literature on large
amplitude dynamic analysis of laminated plates based on higher-order theory.
Moreover, the effect of various plate parameters on the linear and non-linear
fundamental frequency of vibration has not been studied in detail so far. The
non-dimensional frequency of vibration is found to increase with increase in
plate width-to-thickness ratio, both in the linear and the nonlinear range.
The effect of non-linearity is seen significant for plates of width-to-thickness
ratio greater than 40. The in-plane edge conditions have significant influence
on the non-linear frequency of vibration.
Cugnoni et al. (2004) presents efficient C0-compatible
finite elements for modelling laminated composite shells under free vibrations.
Derived from the first-order shear deformation theory (equivalent single-layer
laminate model), the elements are well adapted for evaluating the global dynamic
response (natural frequencies and mode shapes) of moderately thick multilayered
shells. The components of their structural matrices are based on an exact integration
per layer, which results in a higher solution accuracy than with standard explicit
through-the-thickness schemes. The described finite element formulation, which
can be easily implemented in commercial finite element codes, is next validated
by means of several experimental modal test cases on thin to relatively thick
plates or shells.
A discrete method is developed by Huang et al. (2005)
for analyzing the free vibration problem of orthotropic rectangular plates with
variable thickness. The Green function, which is obtained by transforming the
differential equations into integral equations and using numerical integration,
is used to establish the characteristic equation of the free vibration. The
effects of the aspect ratios, boundary conditions and the variation of the thickness
on the frequencies are considered.
For free vibrations of polar orthotropic plate, simple approximate closed form
solutions for mode shapes and its natural frequencies were obtained by Kang
et al. (2005) using the Rayleigh-Ritz method. Coordinate function
satisfying the natural boundary conditions and the predetermined coefficients
was adapted, which results in compact expressions and enables to readily calculate
symmetric and nonsymmetric natural frequencies for arbitrary values of the elastic
constants. The derived formulation can be used in designing of circular plates
such as wood disk, which are naturally endowed with material orthotropy as well
as fiber reinforced composite materials.
A theoretical approach for the free vibration analysis of delaminated unidirectional
sandwich panels is developed by Schwarts-Givli et al.
(2007). The theoretical model accounts for the flexibility of the core in
the out of plane (vertical) direction and the resulting high-order displacement,
acceleration and velocity fields within the core. The analytical approach is
based on Hamiltons variational principle along with the high-order unidirectional
sandwich panel theory and the modified Galerkin method. The two types of models
investigated include delaminated regions with and without contact. The ability
of the model to describe the high-order effects such as the pumping phenomenon
and the localized effects in the vicinity of the delaminated regions is examined.
The state-vector approach is proposed by Chen et al.
(2007) to analyze the free vibration of magneto-electro-elastic laminate
plates. The extended displacements and stresses can be divided into the so-called
in-plane and out-of-plane variables. Once the state equation for the out-of-plane
variables is obtained, a complex boundary value problem is converted into an
equivalent simple initial value problem. Through the state equation, the propagator
matrix between the top and bottom interfaces of every layer can be easily derived.
The global propagator matrix can also be assembled using the continuity conditions.
It is obvious that the order of global propagator matrix is not related to the
number of layers. Consequently, this approach possesses certain virtues including
simple formulation, less expensive computation, etc. To test the formulation,
the developed solution is then applied to a simply supported multilayered plate
constructed of piezoelectric and/or piezomagnetic materials. The natural frequencies
and corresponding mode shapes are computed and compared with existing results.
A mixed layerwise theory and Differential Quadrature (DQ) method (LW-DQ) for
three-dimensional free vibration analysis of arbitrary laminated circular cylindrical
shells is introduced by Malekzadeh et al. (2008).
Using the layerwise theory in conjunction with the three-dimensional form of
Hamiltons principle, the transversely discretized equations of motion
and the related boundary conditions are obtained.
The free vibrations characteristics of simply supported anisotropic composite
laminates are investigated by Ganapathi et al. (2009)
using analytical approach. The formulation is based on the first-order shear
deformation theory and the shear correction factors employed are based on energy
consideration that depends on the layup as well as material properties. The
governing equations are obtained using energy method.
A study of static deformations and free vibrations of shear flexible isotropic
and laminated composite plates with a first-order shear deformation theory is
presented by Ferreira et al. (2009). The analysis
is based on collocation with a Deslaurier Dubuc interpolating basis to produce
highly accurate results. Numerical results for isotropic and symmetric laminated
composite plates are presented and discussed for various thickness-to-length
ratios. The Discrete Singular Convolution (DSC) method for the free vibration
analysis of laminated trapezoidal plates is studied by Murat
et al. (2009). The plate formulation is based on first-order shear
deformation theory (FSDT). The straight-sided trapezoidal domain is mapped into
a square domain in the computational space using a four-node element by using
the geometric transformation. The frequency parameters are obtained for symmetric
angle-ply and cross-ply laminated trapezoidal plate. The accuracy of the present
method is demonstrated by comparing with numerical and analytical solutions
available in the literature.
Amabili and Farhadi (2009) studied (1) the classical
Von Karman theory, (2) the first-order shear deformation theory and (3) the
higher-order (third order) shear deformation theory are compared for studying
the nonlinear forced vibrations of isotropic and laminate composite rectangular
plates. In particular, the harmonic response in the frequency neighborhood of
the fundamental mode of rectangular plates is investigated and the response
curves computed by using the three different theories are compared.
A Ritz approach has been used by Liz and Ricardo (2007)
for the study of the vibration of angle-ply symmetric laminated composite rectangular
plates with edges elastically restrained against rotation and translation. Also,
new numerical results are presented and some results are compared with existing
values in the literature. The free vibration characteristics of laminated composite
and sandwich plates with embedded and/or surface-bonded piezoelectric layers
is studied by Topdar et al. (2007), where a hybrid
plate theory is proposed for modelling the structural system. It involves a
problem of coupled electromechanical field. The variation of mechanical/structural
displacements across the thickness are modelled by an efficient plate theory,
which ensures inter-laminar shear stress continuity as well as stress free condition
at the plate top and bottom surfaces. Levy-type (semi-analytical) finite element
analyses of free vibration and stability of laminated composite rectangular
plates based on both classical and first-order shear deformation theories are
presented by Luccioni and Dong (1998). In this finite
element version, discretization occurs in one coordinate direction (say the
y-axis), leaving the behavior in the x-direction and in time undetermined at
the outset. In this formulation, arbitrary boundary conditions may be imposed
on the two opposite ends of the plate in the y-direction. Hamilton's principle
is used to derive the stiffness, mass and initial stress matrices that enter
into the equations of motion. Periodic solution forms are taken in the x-direction,
whereupon the analyses take the form of algebraic eigen problems from which
the frequencies and critical buckling loads may be extracted.
As the electronic products are desired to have many functions with low weight
and small size increasingly, the ultra-thin and multi-layer Printed Circuit
Boards (PCBs) are required to be used extensively in electronic packaging assemblies.
Usually, these multi-layer PCBs consist of multiple layers of woven glass fiber
reinforced epoxy resin composite substrate sandwiched between copper foils.
The mechanical properties of these multi-layer PCBs can be represented basically
by their bending stiffness. However, complex woven composite material properties
complicate the bending stiffness analysis. In this research, a finite element
analysis model was suggested to describe the bending behavior of woven fiber
composite multi-layer PCB. Both finite element simulation and experiment were
employed in this study by Li et al. (2008).
Analysis of thickness locking in classical, refined and mixed multilayered
plate theories studied by Erasmo Carrera and Brischetto,
(2008). This study has presented a numerical investigation on thickness
locking in classical, refined and advanced mixed theories for one-layered and
multilayered isotropic and orthotropic plates. Closed form solutions have been
given. The following already known conclusions have been confirmed.
||The use σzz = 0 condition appears a suitable
technique to contrast TL in thin-plate analysis. It is very effective to
contrast TL in the case of TPT and FSDT applications for the evaluation
of global parameters such as displacement amplitudes and/or circular frequencies
||σzz = 0 preserves its advantages if applied to plate theories
(HOT) with linear distribution of transverse displacement field uz
||TL appears if and only if a plate theory shows a constant distribution
of transverse normal strain ε zz; that is to avoid TL the plate theories
would require at least a parabolic distribution of transverse displacement
After constructing some elasticity models, a set of close three-dimensional
linear analytical solutions, taking account of all of the normal stresses, shear
stresses and satisfying all the equations of equilibrium, the mid-plane clamped
boundary conditions and interfacial continuity conditions through-thickness,
are presented for axially symmetrical homogeneous isotropic circular plates,
laminates and sandwich plates under uniform transverse load by using the variable-separating
method and formulating a set of displacement functions by Luo
et al. (2004). Reasonability of the present solutions is demonstrated
comparing with FEM analysis, pb-2 Ritz theory analysis and experimental results
of sandwich plates.
Linear static analysis and finite element modeling for laminated composite
plates using third order shear deformation theory is studied by Aagaah
et al. (2003). In this study, deformations of a laminated composite
plate due to mechanical loads are presented. Third order shear deformation theory
of plates, which is categorized in equivalent single layer theories, is used
to derive linear dynamic equations of a rectangular multi-layered composite
plate. Moreover, derivation of equations for FEM and numerical solutions for
displacements and stress distributions of different points of the plate with
a sinusoidal distributed mechanical load for Navier type boundary conditions
BUCKLING AND POSTBUCKLING ANALYSIS
An semi-analytical solution method is presented for the buckling analysis of plates with a reinforced cutout under both uniform and non-uniform compression loading. The cutout is reinforced on both sides of the plate. The solution for buckling response is achieved in two subsequent steps: pre-buckling response from in-plane stress analysis and the critical buckling load and its mode from a buckling analysis that utilizes the prebuckling stresses. The pre-buckling stresses are obtained based on the principle of minimum potential energy while automatically satisfying the equilibrium equations and compatibility condition. Hence, the strain energy of the plate is directly achieved by boundary integrals only. However, there exist material and thickness discontinuities between the unreinforced and reinforced regions of the plate. Unlike the pre-buckling equilibrium equations, the buckling equations, which are obtained from the Treftz criterion, are highly complex and the potential functions automatically satisfying these equations do not exist. Hence, the Treftz criterion is applied with the assumed transverse displacement functions utilizing the full domain integration. In both pre-buckling and buckling analyses, the form of the assumed functions (real or complex) does not necessarily satisfy the kinematic boundary conditions. The kinematic boundary conditions are applied by introducing elastic springs along the boundaries and by enforcing the displacement field to satisfy kinematic boundary conditions through energy minimization.
A global/local analysis methodology for obtaining the detailed stress state
of stepped, square plates with cutouts is presented by Kapania
et al. (1997). The method is based on a global Ritz analysis and
a local finite element analyses. The method is evaluated for an isotropic square
plate with circular and elliptical cutouts and also for stepped plates with
circular cutout. The method was also tested for the composite plate with a circular
Parhi et al. (2001) studied the first ply failure
analysis of laminated composite plates with arbitrarily located multiple delaminations
subjected to transverse static load as well as impact. The theoretical formulation
is based on a simple multiple delamination model. Conventional first order shear
deformation is assumed using eight-noded isoparametric quadratic elements to
develop the finite element analysis procedure. Composite plates are assumed
to contain both single and multiple delaminations. For the case of impact, Newmark
time integration algorithm is employed for solving the time dependent multiple
equations of the plate. Tsai-Wu failure criterion is used to check for failure
of the laminate for both the cases. To investigate the first ply failure, parametric
studies are made for different cases by varying the size and number of delaminations
as well as the stacking sequences and boundary conditions.
Static tests were carried out by Caprino et al.
(2002) on moderately anisotropic, simply supported circular plates made
of graphite fibre reinforced plastic laminates of various thicknesses, loading
them at the centre by a hemispherical tup. A non-linear solution available for
large deflections of isotropic plates was suitably modified to account for the
Hertzian contact phenomena and adopted to model the plate behaviour in the elastic
field. From the experimental results, an unacceptable error is made in predicting
the force-deflection curve up to first failure, if the non-linear portion of
the deflection is neglected. This error is the more evident when thin laminates
are concerned. For thick laminates, the Hertzian contact plays a significant
role in affecting the plate behaviour. Taking into account both the local deformation
and non-linearity due to large deflections, a very accurate prediction of the
load-deflection curve was obtained. Only in the case of the thinnest composites
tested, a divergence of the theoretical curve from the experimental data was
observed at sufficiently high loads. The analysis of the failure modes revealed
that the discrepancy is seemingly attributable to internal damage not resulting
in clearly discernible discontinuities in the load-deflection curve.
The behaviour of laminated composite plates under static loading is studied
by Latheswary et al. (2004b) using a four-noded
element with seven degrees of-freedom per node, based on higher-order shear
deformation theory. The effects of plate width-to-thickness ratio, fibre orientation,
number of layers, thickness ratio, aspect ratio and boundary conditions on the
displacement and stress response of symmetric and anti-symmetric laminated composite
plates subjected to uniformly distributed normal loads are presented. The non-dimensional
central deflection is found to decrease with increase in plate width-to-thickness
ratio. The central deflection approaches a minimum for 45°C fibre orientation.
The number of layers does not have much influence on the central deflection
beyond six layers. The thickness of individual layers plays an important role
in the response of the plate.
Van Phu and Dat (2005) deals with the analysis of non-linear
multilayered reinforced composite plates with simply supported along its four
edges by Bubnov- Galerkin and Finite Element Methods. Numerical results are
presented for illustrating theoretical analysis of reinforced and unreinforced
laminated composite plates.
The high stress concentration at the edge of a cutout is of practical importance
in designing of the engineering structures. There is not any closed form solutions
for a plate with a general shape cutout. These types of cutout usually are determined
either experimentally or numerically using finite element methods. The simple
analytical stress analysis provides a numerical result for stress concentration
factors for perforated plates. This study has presented an analytical solution
for stresses in composite plates with special shaped cutouts. The stress concentration
of isotropic and composite plates with variety of centrally located cutout was
investigated by Rezaeepazhand and Jafari (2005). Analytical
and numerical studies were conducted to investigate the effects of variation
in cutout shape and geometries on the location and the value of the maximum
stress in flat plate under uni-axial tension load. Leknitsjkis solution
for circular and elliptical cutout is extended to special cutout shape using
complex variable mapping. This complex variable function can be used in modeling
and evaluation of stress distribution in perforated composite and isotropic
plates. The stress concentration factor of perforated plates can be significantly
change by using proper material properties and cutout parameters.
The buckling behaviour of composite plates (unidirectional and cross-ply, symmetrical
and non-symmetrical), with multiple delaminations of equal length, subjected
to uniaxial compressive load, has been studied with FEM performing linear and
non-linear analyses. More general cases than those present in the literature
have been analysed. The influence of stacking sequence and of number, position
and length of the delaminations, on the value of the critical load and on threshold
values between global, mixed and local behaviours, has been investigated by
Cappello and Tumino (2006). Linear and non-linear analyses
give similar results when the unstable sublaminates are located in the external
parts of the laminate. When these sublaminates are located internally results
can be different: the discordance could be caused by possible interpenetration
of plies during buckling, that occurs when using a linear analysis, which can
give more conservative results, contrary to what happens when using a non-linear
analysis and in reality.
In general, for a small delamination length (a/L < 0.2) the influence is small and the characteristics at buckling are similar to those of the laminates without cracks (global buckling). The influence increases as the delamination length increases and causes the local buckling of a sublaminate that precedes any other mode of instability because a sublaminate is slender enough when compared with laminate as a whole; in this case significant out-of-plane deflection occurs, generally, in the thinner sublaminates. Regarding to cross-ply laminates, numerical analyses demonstrate that with laminae at 0° on the surface the buckling load of the undamaged composite increases and a high strength to buckling can be obtained also in presence of delaminations at the interfaces with laminae at 90°.
The distributions of stresses and deflection in rectangular isotropic, orthotropic
and laminated composite plates with central circular hole under transverse static
loading have been studied by Mittal and Jain (2008a)
using finite element method. The aim of author is to analyze the effect of D/A
ratio (where, D is hole diameter and A is plate width) upon Stress Concentration
Factor (SCF) and deflection in isotropic, orthotropic and laminated composite
plates under different transverse static loading condition. Analysis has been
done for symmetric and anti-symmetric composite laminates. The results are obtained
for three different boundary conditions. The variations of SCF and deflection
with respect to D/A ratio are presented in graphical form and discussed. The
finite element formulation is carried out in the analysis section of the ANSYS
The Effect of fibre orientation on stress concentration factor in a rectangular
composite laminate with central circular hole under transverse static loading
has been studied by and Mittal and Jain (2008b) using
Finite Element Method. The percent variations in deflection with fibre orientation
are also compared with deflection in laminate without hole. A finite element
study is made for whole analysis of laminate with a central hole under transverse
A finite strip method for non-linear static analysis based on the tangential
stiffness matrix has been developed, Zahari and El-Zafrany
(2008) using the new concept of polynomial finite strip elements, with Mindlin
(first-order shear deformable element) plate-bending theory for composite plates.
A progressive failure algorithm for composite laminates has been successfully
developed for the new finite strip methods using a stress-based failure criterion,
Tsai-Wu. A finite strip analysis programming package which is capable of performing
non-linear progressive damage analysis for composite stiffened plates and shells
has also been developed with Mindlin plate-bending element. Good agreement with
the finite element results has been observed through various test cases, confirming
the accuracy and reliability of the new developed method.
A higher-order layerwise theoretical framework was developed by Theofanis
and Saravanos (2009) for predicting the through-thickness response of thick
composite and sandwich composite plates. Linear, parabolic and cubic distributions
of the in-plane displacements were assumed through the thickness of each discrete
layer, thus, the composite laminate could be modeled using a small number of
discrete layers compared to linear layerwise theories. Interlaminar shear stress
compatibility conditions were explicitly imposed on the discrete layer stiffness
matrices by using a propagating algorithmic procedure, enabling prediction of
interlaminar shear stress at the interfaces between adjacent discrete layers.
Validations of the current theory with 3-D exact solution and a linear layerwise
theory for thick composite and sandwich composite plates were conducted. The
higher-order layerwise theory proved to be robust, since it accurately predicted
parabolic through-thickness distributions of displacements, strains and stresses
at the interfaces and at free edges and efficient, since it used a small number
of discrete layers to model the composite laminate through-thickness. The deviations
observed indicated the effect of constant transverse displacement through-thickness
assumed in the present formulation. Overall, the validations conducted illustrated
the enhanced capabilities of the present higher order layerwise laminate theory,
as well as, its range of applicability.
Based on the two-dimensional theory of elasticity, an accurate solution is
presented by Malekzadeh (2009) for the static analysis
of thick laminated deep circular arches with general boundary conditions. The
formulations are general in the sense that the effects of the variation of arch
curvature across the cross-section, the transverse shear and normal stresses
are included. Fast rates of convergence of the method are demonstrated and its
high accuracy with low computational efforts is exhibited by comparing the results
with existing solutions in the literature. The effects of different parameters
such as opening angle, thickness to-length and ply angle on the normalized stress
and displacement components of the thick laminated arches with different set
of boundary conditions are investigated. The solutions can be used as benchmarks
for other numerical methods and also to clear the accuracy of the classical
theories such as the first order shear deformation theory.
OPTIMIZATION IN COMPOSITE PLATES
The optimal control problem of minimizing the dynamic response of anisotropic
symmetric or antisymmetric composite laminated rectangular plates with various
boundary conditions is presented by Fares et al.
(2002) using various plate theories. The objective of the present control
problem is to minimize the dynamic response of the plate with minimum possible
expenditure of force. The dynamic response of the structure comprises a weight
sum of the control objective (the total vibrational energy) and a penalty functional
of the control force. In addition to the active control, the layer thickness
and the orientation angle of the material fibers are taken as optimization design
variables. The explicit solutions for the optimal force and controlled deflections
are obtained in forms of double series using the Liapunov-Bellman theory. The
effectiveness of the proposed control and the behavior of the controlled structure
are investigated. Various numerical results including the effect of boundary
conditions, number of layers, anisotropy ratio, aspect ratio and side-to-thickness
ratio on the control process for symmetric and antisymmetric laminates are presented.
Mackerle (2004) presented a bibliographical review
of the finite element analyses and simulations of manufacturing processes of
composite materials and their mechanical properties from the theoretical as
well as practical points of view. Topics include: filament winding process;
braiding, weaving and knitting; fiber preforms and resin injection; pultrusion;
compression molding; injection molding; extrusion and other specific manufacturing
processes and processes in general.
A strength-based multiple cutout optimizations in composite plates using fixed
grid finite element method are studied by Liu et al.
(2006). As a typical structural feature, interior cutouts are often indispensable
in meeting such special technical requirements as laying fuel lines and electrical
cables in aircraft wing spars, opening access holes for the service of interior
parts in machine, ventilating the air of tubes and passing the liquid at the
bottom of container. The presence of these holes, however, may significantly
change the stress intensity and structural performance, which could to a certain
extent affect the operational life of the composite structures. For this reason,
one of the foremost design objectives can be to minimize the resulting stress-induced
failure due to the introduction of cutouts. Based on the Tsai-Hill failure criterion
of the first ply, this study presents a newly developed Fixed (FG) Grid Evolutionary
Structural Optimization (ESO) method to explore shape optimization of multiple
cutouts in composite structures. Different design cases with varying number
of cutouts, ply orientations and lay-up configurations are taken into account
in this study. The examples demonstrate that the optimal boundaries produced
by FG ESO are much smoother than those by traditional ESO. The results show
the remarkable effects of different opening numbers and various lay-up configurations
on resulting optimal shapes. The study also provides an in-depth observation
in the interactive influence of the adjacent cutouts on the optimal shapes.
The finite element formulation presented by Naghipour et
al. (2008), which was based on shear deformation theory (FSDT) and predicts
reasonably good results for the laminated plate of mobile bridge deck with different
stacking sequence and fiber orientation. The results have also been compared
with the result of finite difference method, based classic plat theory (CLPT).
This comparison was done to determine the maximum deflection. The principle
normal stress and shear stress of the central point was also determined using
the finite element formulation. The studies reveal the influence of various
parameters and show the following facts:
||The central deflection is a minimum for 0 fiber orientation
||The central deflection decreases with increase in stiffness ratio
||The normal stress and in-plane shear stress with increase in stiffness
||The central deflection decreases with increase in number of layers, but
the rate of decrease is negligible beyond 20 layers
||The normal stress is found to decrease and in-plane shear stress is found
to increase with increase in the number of layers for plate of arrangements
in all cases
||The variation of deflection, normal stress and inplane shear stress with
stiffness ratio follow the same pattern for both simply supported and clamped
conditions. But the variation of transverse shear stress with stiffness
ratio for clamped plates is different from that for simply supported plates
Effect of fibre orientation: Four-layer symmetric and anti-symmetric
laminates with the angle of fibre orientation varying from 0 to 45° with
b/h = 10 and 100 are analysed. A change in fibre orientation angle from 0 to
45° leads to an increase in the fundamental frequency of vibration in the
case of both thick (b/h = 10) and thin (b/h = 100) plates as in Fig.
1. It can also be seen that the fundamental frequency of vibration for symmetric
arrangement is less than that for antisymmetric arrangement, the difference
being more for higher values of α.
Effect of number of layers: The effect of number of layers on non-linear fundamental frequency is studied by considering both thin and thick crossply and angle-ply laminates with anti-symmetric lay-up. The variation of frequency ratio with number of layers is shown in Fig. 2.
||Variation of non-dimensional fundamental frequency with b/h
|| Variation of frequency ratio with number of layers
Even though there is no change in the frequency ratio with increase in number of layers in the case of thick plates, there is a small change from two to four layers in the case of thin plates and afterwards the value remains practically constant. This is in agreement with the earlier observation that two-layer plates behave differently from multi-layered plates.
Linear and non-linear free vibration analysis of laminated composite plates using a finite element model, based on third order shear deformation theory and including Von-Karman nonlinear strains, is presented. The non-dimensional fundamental frequency of vibration is found to increase with increase in width-to-thickness ratio, material anisotropy and angle of fibre orientation. The edge conditions of the plate play an important role in the frequency of vibration of the system. The fundamental frequency of vibration decreases gradually with aspect ratio. The effect of number of layers is found to be insignificant beyond four layers. The frequency ratio increases with increase in width-to-thickness ratio. The Non-linear fundamental frequency of vibration decreases with increase in fibre orientation angle for thin plates, the maximum decrease being for α = 45°. The number of layers does not have any significant influence on non-linear frequency of vibration beyond four-layers. The effect of non-linearity is found to be significant for plates with width-to-thickness ratio greater than 40 aspect ratio. The effect of number of layers is found to be insignificant beyond four layers. The frequency ratio increases with increase in width-to-thickness ratio. The Non-linear fundamental frequency of vibration decreases with increase in fibre orientation angle for thin plates, the maximum decrease being for α = 45°. The number of layers does not have any significant influence on non-linear frequency of vibration beyond four-layers. The effect of non-linearity is found to be significant for plates with width-to-thickness ratio greater than 40.
The recent advances of the finite element analysis of composite laminated plates
based on various lamination theories, with the focus on the free vibration and
dynamics, buckling and postbuckling analysis, geometric nonlinearity and large
deformation analysis and failure and damage analysis of composite laminated
plates, are reviewed in this study. The development of buckling and postbuckling
analysis under material nonlinearity and thermal effects are emphasised and
in the failure analysis, the concentration is especially on the advances of
the first-ply failure analysis. Based on the authors investigation, it
has been found that the research on the following aspects of the composite laminated
plates is relatively limited and may attract more interests in the future research.
||Material nonlinearity effects on structural behaviour of composite
||Failure and damage analysis under viscoelastic effects such as thermal
and creep effects
||Failure and damage analysis under cyclic loading
||Micromechanical approach for damage analysis
||Analysis of the damage evolution in composite laminates
||Multiscale modelling of crack initiation, propagation and overall structural
||The free vibration problem of antisymmetric laminated plates having translational
as well as rotational edge constraints which appears to have not been studied